Rigorous mathematical modelling for a Fast Corrector Power Supply in TPS

To enhance the stability of beam orbit, a Fast Orbit Feedback System (FOFB) eliminating undesired disturbances was installed and tested in the 3rd generation synchrotron light source of Taiwan Photon Source (TPS) of National Synchrotron Radiation Research Center (NSRRC). The effectiveness of the FOFB greatly depends on the output performance of Fast Corrector Power Supply (FCPS); therefore, the design and implementation of an accurate FCPS is essential. A rigorous mathematical modelling is very useful to shorten design time and improve design performance of a FCPS. A rigorous mathematical modelling derived by the state-space averaging method for a FCPS in the FOFB of TPS composed of a full-bridge topology is therefore proposed in this paper. The MATLAB/SIMULINK software is used to construct the proposed mathematical modelling and to conduct the simulations of the FCPS. Simulations for the effects of the different resolutions of ADC on the output accuracy of the FCPS are investigated. A FCPS prototype is realized to demonstrate the effectiveness of the proposed rigorous mathematical modelling for the FCPS. Simulation and experimental results show that the proposed mathematical modelling is helpful for selecting the appropriate components to meet the accuracy requirements of a FCPS.


Introduction
For advanced scientific investigations in the related fields of molecular, material and medicine etc., the 3 rd generation synchrotron light source of Taiwan Photon Source (TPS) in National Synchrotron Radiation Research Center (NSRRC), a 3 GeV synchrotron facility with very low emittance and ultra-brightness beam source, was constructed and operated since 2014. TPS is a concentric ring where booster and storage rings are built in the same tunnel. The booster ring is used to ramp the electron beam with 150 MeV from a linear accelerator up to 3 GeV at a 3 Hz repetition rate. The storage ring composed of 24 double-bend cells with 6-fold symmetry in a circumference of 518.4 m is then used to extract the electron beam. Due to the small beam sizes both in vertical and horizontal planes, the beam position control has to be in the order of tens of µm and therefore the requirement for the accuracy and stability of beam orbit is high. In order to enhance the accuracy and stability of beam orbit, a Fast Orbit Feedback System (FOFB) composed of corrector magnets, power supplies, beam position data and orbit compensation algorithm was installed and tested in TPS since 2015. The FOFB system can be used to eliminate undesired disturbances coming from magnetic strength errors, alignment errors and water pipe vibrations close to magnets etc. and to suppress the beam orbit instability [1,2].
The effectiveness of the FOFB greatly depends on the output performance of Fast Corrector Power Supply (FCPS) for corrector magnet; therefore, the design and implementation of a high accurate FCPS is essential. To achieve high accuracy and stability for a FCPS, some optimal design issues such as the selection of circuit components, the adjustment of controller parameters, the influences of uncertainties in the current sensor and circuit components on output stability and so on should be considered and integrated into a FCPS design. In general, the commonly-used circuit simulation software including PSCAD, Spice, and PSIM etc. can be used for transient-behavior simulation of a FCPS; however, the software cannot be straightforwardly used to select the circuit components, adjust the parameters of controller and especially revolve the effects of uncertainties in the current sensor and circuit components on output stability. The design of a high accurate FCPS is therefore very time-consuming since many factors need to be premeditated and investigated. A -1 -rigorous mathematical modelling is indispensable for shortening design time and improving design performance of a FCPS [3,4].
A rigorous mathematical modelling derived by the state-space averaging method for a FCPS in the FOFB of TPS composed by a full-bridge topology is proposed in this paper. The MATLAB/SIMULINK software is used to construct the proposed mathematical modelling and to conduct the simulations of the proposed FCPS. The proposed mathematical modelling for the FCPS can be used to select the appropriate components to meet the accuracy requirements of the FCPS. As a case study, the simulations for the effects of different resolutions (bits) of ADC on the output accuracy of the FCPS in the FOFB of TPS are investigated. Simulation results demonstrate the effectiveness and efficiency of using the proposed mathematical modelling for the optimal design of the FCPS. Figure 1 shows the basic circuit configuration of the FCPS in the FOFB of TPS designed and developed by the power supply group of NSRRC. The ±10 A/±50 V FCPS employs the full-bridge topology where V DC is the input DC voltage; S 1 to S 4 are the MOSFETs; L f 1 , L f 2 and C f constitute a low-pass filter; L M and R M are the equivalent load of a corrector magnet; and Shunt is a high precision shunt resistor used as current sensor. A Hybrid Pulse Width Modulation (HPWM) as proposed in [5] requiring two of the four MOSFETs in the full-bridge circuit to be operated at high frequency and the other two be switched at the output frequency is adopted in this paper. Since only two of the four MOSFETs are switched at high frequency, the switching loss can be reduced effectively. The HPWM was designed based on the conventional unipolar PWM which compares a sinusoidal reference signal with a triangular carrier signal to generate the PWMs for MOSFETs. Figure 2 illustrates the theoretical PWM control signals for the MOSFETs S 1 to S 4 and the bridge output voltage assuming that S 1 and S 2 are operated at output frequency and S 3 and S 4 are switched at high frequency. In figure 2, V sin and V sin_inv are the sinusoidal reference signals for the positive and negative bridge output voltages, respectively; V tri is the triangular carrier signal; PWM S1 to PWM S4 are the corresponding PWM control signals for MOSFETs S 1 to S 4 ; and V ab is the bridge output voltage.  It can be observed from figure 2 that PWM S2 and PWM S4 are complementary to PWM S1 and PWM S3 . There are four operational modes in the FCPS. Figure 3 shows the equivalent circuits of modes 1 and 2 for the positive bridge output voltage. As shown in figures 2 and 3(a), mode 1 represents that V sin is larger than 0 and V sin is larger than V tri with PWM S1 , PWM S2 , PWM S3 and PWM S3 being in the ON, OFF, OFF and ON states, respectively, and the bridge output voltage V ab being +V DC . Similarly, mode 2 denotes that V sin is smaller than 0 and V sin is larger than V tri demonstrated in figures 2 and 3(b), with PWM S1 , PWM S2 , PWM S3 and PWM S3 being in the ON, OFF, ON and OFF states, respectively, and the bridge output voltage V ab being 0. Modes 3 and 4 which are similar to modes 2 and 1 belong to the negative bridge output and the details are abridged.  -3 -

Rigorous mathematical modelling of a FCPS
To achieve high accuracy and stability in the FCPS, a rigorous mathematical modelling is derived in this paper to simulate some optimal design issues for the FCPS and therefore to shorten design time and improve design performance. The state-space averaging method that has been widely applied to model power electronic circuits is used in this paper. The state-space averaging method uses the state variables to derive the state equation for the equivalent circuit of each switching mode and then average the state equations using duty cycle to obtain the state-space averaging equation. The transfer function aiming for the stability analysis and optimal parameters design of controller can be solved by the derived state-space averaging equation.   Figure 4 shows the equivalent circuits of modes 1 and 2 where r f L1 and r f L2 are the DC impedances for L f 1 and L f 2 ; r C f is the equivalent series resistance of C f ; r DS is the drain-source resistance of a MOSFET; and R SH is the shunt resistor of the current sensor. For simplicity, the summations of L f 1 and L f 2 and of r f L1 , r f L2 , r DS and r DS are denoted as L T and R T , respectively. From figure 4, it can be observed that the state variables for the FCPS are i LT , i L M , and v C f . Using Kirchhoff circuit laws, eq. (3.1) can be derived.
The output voltage can be expressed as The state equation for mode 1 can be rearranged from (3.1) and (3.2) and expressed as 3) can be rewritten as Using the similar procedures as derived in (3.1) to (3.3), the state equation for mode 2 as shown in figure 4(b) can be expressed as Let T S be one switching period and dT S and d T S be the turn-on time and turn-off time for modes 1 and 2 in the switching period as illustrated figure 4, respectively, the state equations for mode 1 and mode 2 as expressed in (3.4a) and (3.5a) can then be rewritten approximately as (3.6) and (3.7), respectively.
The state equation for one switching period can be obtained approximately by averaging the state equations for modes 1 and 2 expressed in (3.6) and (3.7), they are dT S y 1 +d T S y 2 T S = dC 1 x+dD 1 u+d C 2 x+d D 2 u (3.8b) -5 -Therefore, the state-space averaging equation can be written as The stat-space averaging equations obtained by (3.9) can be integrated into optimal design of the FCPS. Figure 5 illustrates the simulation architecture for the FCPS using state-space averaging equation where i MEA L M is the current signal acquired from the shunt resistor and signal acquisition module; i ADC L M is the digital current signal obtained by the ADC module; and i COM L M is the current command used for Proportional-Integral (PI) controller. From figure 5, it can be observed that the effects of the DC impedances of L f 1 and L f 2 , the equivalent series resistance of C f , the drain-source resistance of a MOSFET and the shunt resistor of the current sensor on the output performance of the FCPS can be effectively analyzed in the proposed simulation architecture. Besides, the uncertainties in the signal acquisition module and ADC module can also be investigated. Therefore, some optimal design issues such as the selection of circuit components, the adjustment of controller parameters, the effects of uncertainties in the current/voltage sensors and circuit components on the output stability and so on can be considered and integrated into the FCPS design using the derived state-space averaging equation and the proposed simulation architecture.

Simulation and experimental results
The MATLAB/SIMULINK software is used to construct the proposed mathematical modelling and to conduct the simulations of the FCPS. Figure 6 shows the simulation block diagram constructed by MATLAB/SIMULINK software. Table 1 lists the circuit parameters used in the designed FCPS. The proposed simulation architecture can be used to design and select the components for the deigned FCPS. As a case, the simulations for the effects of the different resolutions of ADC on the output accuracy of the proposed FCPS are investigated. The rated output current i L M is ±10 A.

JINST 12 T04004
The output current is transformed to an input voltage in ±0.1 V for the signal acquisition module by the shunt resistor. The output current signal i MEA L M is in ±10 V and the ADC module with the input range of ±12 V is used to convert the analog output current signal i MEA L M to the digital current signal i ADC L M . The accuracy requirement for the output current i L M is that the mean error must be lower than 10 ppm (100 uA) at 10 A.
The maximum round-off error for an ADC module with the input range of ±12 V can be calculated by where N is the number of bits for the ADC module. The 12, 14, 16, 18 and 20-bit ADC modules are simulated by the proposed architecture. The maximum round-off errors for the different bits are listed in table 2. From (4.1) and table 2, the digital values in integer format, the floating values (V) and the round-off errors (V) for input value 10 V converted by the 12, 14, 16, 18 and 20-bit ADC modules are shown in table 3. Obviously, the larger number of bit for the ADC module has lower round-off error; however, it causes greater computational burden in the controller performance. Therefore, the appropriate number of bit for the ADC module needs to be designed thoughtfully. The proposed simulation architecture as shown in figure 6 can effectively take the influences of different bits of the ADC module on the output current accuracy into account. Besides, the components and parameters in the FCPS such as PI parameters in the controller also affect the output current accuracy and stability, thus needing to be considered together.  Table 4 shows the performances of different bits of ADC module for the accuracy and stability of the designed FCPS at rated output current 10 A where mean, standard, min and max denote the mean, standard deviation, minimum and maximum values of the output current, respectively and the mean error indicates mean error value of output current. In table 4, it can be observed that the 16, 18 and 20-bit ADC modules can meet the accuracy requirement of the mean error of output current being lower than 10 ppm (100 uA) at 10 A. Comparing table 4 with tables 2 and 3, it can also be seen that the 16-bit ADC module has larger round-off error; however, the mean error obtained from the simulations is less than 10 ppm. Meanwhile, the 18-bit ADC module has about 4 times larger round-off error than that of the 20-bit ADC module; however, the simulated mean errors for -7 -  the 18 and 20-bit ADC modules are close. The computational performance of controller and PI parameters used in the controller may be the main reasons resulting in the differences. Figures 7-9 illustrate the simulated output current spectra for the 16, 18 and 20-bit ADC modules, respectively. From figures 7-9, it can be seen that the output current errors for the 16, 18 and 20-bit ADC module from 0 Hz to 1.6 kHz are lower than 10 ppm. Since the performance of the 18-bit ADC module is close to the 20-bit ADC module, the 18-bit ADC module is selected and used in the designed FCPS. Figure 10 shows the realized FCPS prototype using the 18-bit TI ADS8382. Figure 11 illustrates the measured output current spectrum from 0Hz to 1.6 kHz. From figure 11, it can be seen that the output current error of realized FCPS prototype is far less than 10 ppm; therefore, the effectiveness of the proposed rigorous mathematical modelling for the FCPS can be proved.

Conclusions
The effectiveness of the FOFB greatly depends on the output performance of FCPSs. A rigorous mathematical modelling is very useful to shorten design time and improve design performance of a FCPS and therefore enhance the stability of beam orbit. By using the state-space averaging method, a rigorous mathematical modelling of the full-bridge FCPS in the FOFB of TPS was derived in this paper. The MATLAB/SIMULINK software was used to construct the proposed mathematical modelling and to conduct the simulations of the designed FCPS. The influence of different resolutions of ADC on the output accuracy of the FCPS was investigated. Simulation results showed that the proposed mathematical modelling is helpful for selecting the appropriate components to meet the accuracy requirements of the FCPS. Other optimal design issues such as the selection of circuit components, the adjustment of controller parameters, the influence of uncertainties in the current/voltage sensors and circuit components on the stability of FCPS and so on will be investigated in the future research.